{
  "id": "2308.10096",
  "version": "v1",
  "published": "2023-08-19T19:48:47.000Z",
  "updated": "2023-08-19T19:48:47.000Z",
  "title": "Essential dimension of symmetric groups in prime characteristic",
  "authors": [
    "Oakley Edens",
    "Zinovy Reichstein"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "The essential dimension $\\operatorname{ed}_k({\\rm S}_n)$ of the symmetric group ${\\rm S}_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1 x^{n-1} + \\ldots + a_n$ can be reduced to a $d$-parameter form by a Tschirnhaus transformation. Finding this number is a long-standing open problem, originating in the work of Felix Klein, long before essential dimension was formally defined. We now know that $\\operatorname{ed}_k({\\rm S}_n)$ lies between $\\lfloor n/2 \\rfloor$ and $n-3$ for every $n \\geqslant 5$ and every field $k$ of characteristic different from $2$. Moreover, if $\\operatorname{char}(k) = 0$, then $\\operatorname{ed}_k({\\rm S}_n) \\geqslant \\lfloor (n+1)/2 \\rfloor$ for any $n \\geqslant 6$. The value of $\\operatorname{ed}_k({\\rm S}_n)$ is not known for any $n \\geqslant 8$ and any field $k$, though it is widely believed that $\\operatorname{ed}_k({\\rm S}_n)$ should be $n-3$ for every $n \\geqslant 5$, at least in characteristic $0$. In this paper we show that for every odd prime $p$ there are infinitely many positive integers $n$ such that $\\operatorname{ed}_{\\mathbb F_p}(\\rm{S}_n) \\leqslant n-4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-19T19:48:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12E05",
      "14G17",
      "14L30",
      "14E05"
    ],
    "keywords": [
      "essential dimension",
      "symmetric group",
      "prime characteristic",
      "odd prime",
      "minimal integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}